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P.Ravindran, Computational Condensed Matter Physics, Auguest 2015 Basics for ab initio Calculations
Basics for ab initio Calculations
http://folk.uio.no/ravi/CMT2015/
Prof.P. Ravindran, Department of Physics, Central University of Tamil
Nadu, India
P.Ravindran, Computational Condensed Matter Physics, Auguest 2015 Basics for ab initio Calculations
Approximation #1: Separate the electrons into 2 types:
Core Electrons & Valence Electrons
The Core Electrons: Those in the filled, inner shells of the
atoms. They play NO role in determining the electronic properties
of the solid.
Example: The Si free atom electronic configuration:
1s22s22p63s23p2
Core Shell Electrons = 1s22s22p6 (filled shells)
These are localized around the nuclei & play NO role in the
bonding.
Lump the core shells together with the Nuclei Ions
(in ∑i , include only the valence electrons)
Core Shells + Nucleus Ion Core
He-n He-i , Hn Hi
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The Valence Electrons
Those in the unfilled, outer shells of the free atoms. These determine the electronic properties of the solid and take part in the bonding.
Example:
The Si free atom electron configuration:
1s22s22p63s23p2
The Valence Electrons = 3s23p2 (unfilled shell) In the solid, these hybridize with the electrons on neighbor
atoms. This forms strong covalent bonds with the 4 Si nearest-neighbors in the Si lattice
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Methods for Efficient Computation
K-points (k)
– Discrete points specified in Brillouin Zone used to perform numerical integration during calculation.
Energy Cut-off Value (ecut)
– Energy value for maximum energy state included in a summation over electron states.
Pseudopotentials (pp)
– Offers specific exchange-correlation functional form which represents frozen “core electrons.” They are based largely on empirical data.
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LDA vs. GGA Approximations
“Local-density approximations (LDA) are a class of
approximations to the exchange-correlation (XC) energy
functional in DFT that depend solely upon the value of the
electronic density at each point in space (and not, for example,
derivatives of the density or the Kohn-Sham orbitals).” This is
more of a first-order approximation.
“Generalized gradient approximations (GGA) are still local but
also take into account the gradient of the density at the same
coordinate.”
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Ensuring k and Ecut Lead to a Converged Energy
The most important skill in performing DFT calculations is the
ability to get converged energies. Since the appropriate choice of
k-points and Ecut vary wildly among different geometries (and
even different required accuracies), it is important to be able to
form the following graphs every time you perform ‘scf’
calculations on new geometries.
Converged values of Ecut and k should be reported any time you
publish DFT results, so that someone else may reproduce your
calculation and agree on the same numerical error.
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Energy Cut-off Convergence Plot
Not only should the graph look converged, but the difference in energy between
the last two consecutive points should be smaller or equal to your required accuracy!
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K-point Convergence Plot
As we can see from the
convergence plots, the
presence of smearing does
little to ensure convergence
with fewer k-points.
Note: In an automatic
distribution of k-points, the value
of k specifies how many discrete
points there are equally-spaced
along each lattice vector to
populate the Brillouin Zone.
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K-point Convergence Plot (cont.)
When unit cells do not have equal-length lattice vectors, it is sometimes computationally rewarding to “geometrically-optimize” your automatic k-point distribution.
For example, if one had a unit cell that was four times taller in one direction than its other two directions, one should specify only a quarter as many k-points along the taller direction.
– This makes sense, because in reciprocal space, the taller distance will only be a quarter as long as the other two distances.
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Comparing the Relaxed Structure to Literature
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Comparing the Relaxed Structure to Literature (cont.)
On top of comparing the ‘a’ and ‘c/a’ lattice parameters to literature and to experiment, it is also useful to compare a quantity called “cohesive energy”
– The difference between the average energy of the atoms of a crystal and that of the free atoms.
It is important to compare cohesive energies and not specifically the self-consistent energies, because only differences in energies are physically meaningful.
– The energy datum for an ‘scf’ calculation is specified by the choice of pseudopotential.
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Unit Cell: the smallest group of atoms possessing the
system of the crystal which, when repeated in all
directions, will develop the crystal lattice.
Primitive Cell: is a type of minimum-volume cell that
will fill all space by the repetition of suitable translation
operation.
Primitive Cell
Unit Cell
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Bravais Lattice and Crystal structures
The Fourteen Space (Bravais) Lattice : well known
P - primitive cell;
I - body-centered cell;
F - face-centered cell,
C - Centered: additional point in the center of each end
R - Rhombohedral: Hexagonal class only
Monoclinic and
Triclinic Cells
Orthorhombic Cells
p
FC
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Hexagonal and
Triagonal Cells
Tetragonal Cells
Cubic Cells
Rhombohedral
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Rotation
Reflection
Inversion
32 point groups
Translation
14 Space Lattices
Screw
Glide
230 Space Groups
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▲Miller Indices:
Indices for Direction: [x,y,z] represents a specific
crystallographic direction; <x,y,z> represents all
of the directions of the same form
Indices for Planes: (x,y,z) represents a specific plane;
{x,y,z} represents all of the planes of the same
form [x,y,z] is the directional normal of plane
(x,y,z)
Miller Indices for Hexagonal Crystals: 4-digit system
(hklm); h+k=-l => reduced to 3-digit system
a1
a2
a3
c hkl: reciprocals of the intercepts with
a1, a2, and a3, m reciprocals of the
intercepts with c.
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Miller Indices (Example)
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Reciprocal Lattices and the First Brillouin Zone
Reciprocal Lattice
– Crystal is a periodic array of lattices
Performing a spatial Fourier transform
– Reciprocal Lattice
Expression of crystal lattice in fourier space
t
Reciprocal lattice
FT for time
FT for space
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Reciprocal Lattice
*a
b-c plane
Next lattice plane
d*aa
* b ca
a b c
* 1| |a
d
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Defined as a Wigner-Seitz unit cell in the reciprocal lattice
The waves whose wavevector starts from origin, and ends at this
plane satisfies diffraction condition
Brillouin construction exhibits all the
wavevectors which can be Bragg
reflected by the crystal
k
Wigner-Seitz unit cell
22k G G
Brillouin zone
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First Brillouin Zone
The smallest of a Wigner-Seitz cell in the reciprocal lattice
The reciprocal lattices
(dots) and corresponding
first Brillouin zones of
(a) square lattice and
(b) hexagonal lattice.
The first brillouin zone is the
smallest volume entirely enclosed
by planes that are the
perpendicular bisection of the
reciprocal lattice vectors drawn
from the origin
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The Brillouin zone (BZ)
Irreducible BZ (IBZ)
– The irreducible wedge
– Region, from which the whole BZ can be obtained by applying all symmetry operations
Bilbao Crystallographic Server:
– www.cryst.ehu.es/cryst/
– The IBZ of all space groups can be obtained from this server
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Wigner-Seitz Cell
Form connection to all neighbors and span a plane normalto the connecting line at half distance
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Wigner-Seitz Cell & Brillouin Zone
1st Brillouin Zone is the Wigner-Seitz Cell of the reciprocal lattice
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BCC WS cell BCC BZ
FCC WS cell FCC BZ
Real Space Reciprocal Space
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29The first Brillouin zone of a FCC structure
Face-centered cubic
KMiddle of an edge joining two hexagonal faces
L Center of a hexagonal face
UMiddle of an edge joining a hexagonal and a square face
W Corner point
X Center of a square face
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Concepts when solving Schrödingers-equation
k
i
k
i
k
irV
)(
2
1 2
non relativistic
semi-relativistic
fully-relativistic
“Muffin-tin” MT
atomic sphere approximation (ASA)
Full potential : FP
pseudopotential (PP)
Hartree-Fock (+correlations)
Density functional theory (DFT)
Local density approximation (LDA)
Generalized gradient approximation (GGA)
Beyond LDA: e.g. LDA+U
Non-spinpolarized
Spin polarized
(with certain magnetic order)
non periodic
(cluster)
periodic
(unit cell)
plane waves : PW
augmented plane waves : APW
atomic orbitals. e.g. Slater (STO), Gaussians (GTO),
LMTO, numerical basis
Basis functions
Treatment of
spin
Representation
of solid
Form of
potential
exchange and correlation potential
Relativistic treatment
of the electrons
Schrödinger - equation
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Muffin-Tin Potential
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P.Ravindran, Computational Condensed Matter Physics, Auguest 2015 Basics for ab initio Calculations
K-point sampling
where the wave vector k is located in the first Brillouin zone (BZ).
Bloch’s theorem states that the wave function in a periodic
crystal can be described as:
It is therefore necessary to sample the wave
function at multiple k-points in BZ to get a
correct description of the electron density
and effective potential.
Using symmetry lowers the number of
necessary k-point to the ones in the
Irreducible Brillouin zone (IBZ).
IBZ
kx
ky
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1. In a periodic solid:
Number of atoms
Number and electrons
Number of wave functions ??
Bloch theorem will rescue us!!
2. Wave function will be extended over the entire solid
Periodic systems are idealizations of real systemsComputational problems
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A periodic potential commensurate with the lattice. The Bloch theorem
Bloch Theorem: The eigenstates of the one-electron Hamiltonian in a
periodic potential can be chosen to have the form of a plane wave times a
function with the periodicity of the Bravais lattice.
Periodicity in reciprocal space
Reciprocal lattice vector
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The wave vector k and the band index n allow us to label each electron (good quantum numbers)
The Bloch theorem changes the problem
Instead of computing an infinite
number of electronic wave functionsFinite number of wave functions at an
infinite number of k-points in the 1BZ
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Many magnitudes require integration of Bloch functions over Brillouin zone (1BZ)
Charge density
In practice: electronic wave functions at k-points that are very close
together will be almost identical
It is possible to represent electronic wave functions over a region of
k-space by the wave function at a single k-point.
Band structure energy
In principle: we should know the eigenvalues and/or eigenvectors at
all the k-points in the first BZ
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Recipes to compute sets of spetial k-points for thedifferent symmetries to accelarate the convergence of BZ integrations
Baldereschi (1973)
Chadi and Cohen (1973)
Monkhorst-Pack (1976)
The magnitude of the error introduced by sampling the Brillouin zone with a
finite number of k-points can always be reduced by using a denser set of points
P.Ravindran, Computational Condensed Matter Physics, Auguest 2015 Basics for ab initio Calculations
11 papers published in APS journals since 1893 with >1000 citations in
APS journals (~5 times as many references in all science journals)
From Physics Today, June, 2005
The number of citations allow us to gauge the importance of the works on DFT
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k-samplingMany magnitudes require integration of Bloch
functions over Brillouin zone (BZ)
r dk n k BZ
i
i k 2
In practice: integral sum over a finite uniform grid
sampling in k is essential Essential for:
Small systems
Real space Reciprocal space
Metals Magnetic systems
Good description of the Bloch
states at the Fermi level
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Sampling of the k-points
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K-point sampling in the BZ
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Linear Interpolation of eigen values to improve the BZ
integration
Brillouin Zone Integration – Tetrahedrons method
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Temperature Effects
“Occupancy” of state =
PE,T = Probability that state at energy E will be occupied at
temperature T
= f(E) (“Fermi function)
k=Boltzmann constant = 8.62 × 10-5 eV/K
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0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
0.2
0.4
0.6
0.8
1
1.2
E/Ef
f(E
) T=0
T=300K
T=600K
Fermi Function vs. T
Ef=1 eV
kT=26 meV (300K)
kT=52 meV (600K)
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Density of states (DOS)
Method:
Calculate the Kohn-Sham eigenvalues with a very
dense k-point mesh.
Use a Gaussian or Lorentzian broadening function
for the delta function.
Perform the summation of the states over the
Brillouin zone.
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Projected density of states (PDOS)
Method:
Calculate the Kohn-Sham eigenvalues i and wave
functions i.
Calculate the overlap between the Kohn-Sham wave
functions i and atomic wave functions fal
Use a Gaussian or Lorentzian broadening function
for the delta function.
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Band Structure & DOS for Graphene
DO
S
K G M K G -F [eV]
- F
[eV
]
k[Å-1] van Hove singularities
K
MG
Brillouin Zone
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PDOS for Graphene
DO
S
-F [eV]
PD
OS
K
MG
Brillouin Zone
-F [eV]
pz
px,py
s
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DOS of real materials: Silicon, Aluminum, Silver
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Measuring DOS: Photoemission spectroscopy
1( ) ( ) ( ) ( ) ( )
( )kin bin bin kin kin
kin
I D f D f f f
Fermi Golden Rule: Probability per
unit time of an electron being ejected
is proportional to the DOS of occupied
electronic states times the probability
(Fermi function) that the state is
occupied:
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Measuring DOS: Photoemission spectroscopy
Once the background is subtracted
off, the subtracted data is proportional
to electronic density of states
convolved with a Fermi functions.
We can also learn about DOS above the Fermi
surface using Inverse Photoemission where electron
beam is focused on the surface and the outgoing flux
of photons is measured.
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Band Filling Concept
Electron bands determined by lattice and ion core potentials
Bands are filled by available conduction and valence electrons
Pauli principle → only one electron of each spin in each (2π)3 volume in “k-space”
Bands filled up to “Fermi energy”
– Fermi energy in band → metal
– Fermi energy in gap → insulator/semiconductor
For T≠0, electron thermal energy distribution = “Fermi function”
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Energy bands
The inter-atomic spacing varies with the direction you are
moving in the crystal {111}planes would have a smaller d
than {100} planes, thus the energy gaps are different in
each direction.
This gives rise to different bands, but usually effect due to
this can be averaged.
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1D case
3D case
- Band structure Indirect gap
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Direct gap
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“Density of States”: N(E)
N(E) dE = number of electron states between E and E+dE
n = number of electrons per unit volume
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Isotropic Parabolic Band
k
E
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Density of States:
Isotropic, Parabolic Bands
NT(E) = number of states/unit volume with energy<E
= “k-space” volume/(2π)3 (per spin direction)
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Electron Density of States: Free Electrons
2
22 2
2
mEk
m
kE
3/2
3 2 2
1 1 2( )
2
dN mD E E
L dE
3 23 3 3
2 2 2
2
3 3
/k L L mE
N
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Electron Density of States: Free Electrons
D(E)
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D(ε)
D
Density of States (DOS)
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81Bandstructure and DOS
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Electronic Structure and Chemical BondingJ.K. Burdett, Chemical Bonding in Solids
(eV)
-14
-12
-10
-8
-6
-4
-2
0
2
4
6
8
10
L G X W L K G
Electronic Structure of Si:
Fermi Level
Electronic Band Structure Electronic Density of States
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Real Band Effects
Thermal Eg ≠ Optical Eg
Electron effective mass ≠ Hole effective mass
More than one electron/hole band
– Multiple “pockets”
– Overlapping bands
Anisotropic electron/hole pockets
Non-parabolic bands
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Real Band Effects
Eg
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85LDA vs. LSDA
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What we can calculate
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Murnaghan Equation of State is usually used to extract Bulk
modulus from Total energy curve
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What About the Kohn-Sham Eigenvalues?
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Attempts on Improving LDA
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Slab structure of Al(110)
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Surface Relaxation and Reconstruction
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Surface Relaxation
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Surface Relaxation
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Relaxation of Pt (210)
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Surface Reconstruction: Missing Row Reconstruction
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Semiconductor Surfaces: Si(100)